If $$ \sin A = \frac{1}{x} $$ then $$ \cos A = ? $$

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RRB NTPC Exam 27 Jan 2021 click here to view all questions

If $$ \sin A = \frac{1}{x} $$ then $$ \cos A = ? $$

A$$ \frac{\sqrt{x^2+1}}{x} $$

B$$ \frac{x}{\sqrt{x^2-1}} $$

C$$ \frac{\sqrt{x^2-1}}{x} $$

D$$ \frac{x}{\sqrt{x^2+1}} $$

Akhilesh ? Apr 1 '25 at 20:40

correct answer is: option c [\( \frac{\sqrt{x^2-1}}{x} \)]
Explanation: We know that \( \sin^2 A + \cos^2 A = 1 \). Therefore, \( \cos^2 A = 1 - \sin^2 A \). Given that \( \sin A = \frac{1}{x} \), we can substitute this into the equation to get \( \cos^2 A = 1 - (\frac{1}{x})^2 = 1 - \frac{1}{x^2} = \frac{x^2 - 1}{x^2} \). Taking the square root of both sides, we get \( \cos A = \sqrt{\frac{x^2 - 1}{x^2}} = \frac{\sqrt{x^2 - 1}}{x} \).

• Option a: \( \frac{\sqrt{x^2+1}}{x} \) is incorrect because it uses addition instead of subtraction under the square root. It incorrectly calculates \( \cos A \) using a flawed manipulation of the Pythagorean identity.
• Option b: \( \frac{x}{\sqrt{x^2-1}} \) is incorrect because it is the reciprocal of the correct answer, and incorrectly places x in the numerator and the root in the denominator.
• Option c: \( \frac{\sqrt{x^2-1}}{x} \) is the correct answer, as shown in the explanation.
• Option d: \( \frac{x}{\sqrt{x^2+1}} \) is incorrect for two reasons: it contains a '+' instead of a '-', and the numerator and denominator are swapped from the correct response following algebraic manipulation.

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correct answer is: option c [\( \frac{\sqrt{x^2-1}}{x} \)] Explanation: We know that \( \sin^2 A + \cos^2 A = 1 \). Therefore, \( \cos^2 A = 1 - \sin^2 A \). Given that \( \sin A = \frac{1}{x} \), we can substitute this into the equation to get \( \cos^2 A = 1 - (\frac{1}{x})^2 = 1 - \frac{1}{x^2} = \frac{x^2 - 1}{x^2} \). Taking the square root of both sides, we get \( \cos A = \sqrt{\frac{x^2 - 1}{x^2}} = \frac{\sqrt{x^2 - 1}}{x} \). Option a: \( \frac{\sqrt{x^2+1}}{x} \) is incorrect because it uses addition instead of subtraction under the square root. It incorrectly calculates \( \cos A \) using a flawed manipulation of the Pythagorean identity. Option b: \( \frac{x}{\sqrt{x^2-1}} \) is incorrect because it is the reciprocal of the correct answer, and incorrectly places x in the numerator and the root in the denominator. Option c: \( \frac{\sqrt{x^2-1}}{x} \) is the correct answer, as shown in the explanation. Option d: \( \frac{x}{\sqrt{x^2+1}} \) is incorrect for two reasons: it contains a '+' instead of a '-', and the numerator and denominator are swapped from the correct response following algebraic manipulation. Akhilesh ? answered Apr 1 '25 at 20:40

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